Let $V$ be a Banach space and $B(V)$ be the bounded operators on $V$. Now, the space $B(V)$ has a norm, the strong topology (which is the topology of pointwise convergence) and it has a stronger topology called ultrastrong or $\sigma$-strong topology. It is known that these two topologies coincide on norm-bounded sets. I want to know if the $\sigma$-strong topology is the strongest topology which coincides with the strong topology (respectively itself) on bounded sets.

Edit:
The answer seems to be "no" for this general case. I have no example though.
To simplify this question, I removed a second part in which I asked for a special case and will reformulate it as its own question. Thank you for the helpful comments on both parts.